REVIEW
On sums of two primes and six Fibonacci numbers
Number Theory · Submitted 18 Aug 2026 · 23 references
Pith's verdict: New asymptotic-basis results for primes with Fibonacci/Lucas numbers, likely correct but not fully verifiable from the manuscript because numerical certificates are asserted, not shipped.
T0 · AI review
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Referee report
The paper
The paper claims
This paper establishes that every sufficiently large positive integer is a sum of two primes and six positive Fibonacci numbers, with analogous Lucas and mixed versions.
The reading
What carries the argument
The load-bearing object is the weighted representation integral with forced parity, such as $R_e(N)$ for even $N$ and $R_o(N)=\int_0^1 S(\alpha)^2A(\alpha)B(\alpha)^5e(-N\alpha)\,d\alpha$ for odd $N$. For the Lucas and mixed cases the paper uses the parity projectors $P_{L,7,N}(\alpha)=\frac12(T_L(\alpha)^7+(-1)^NT_L^\sharp(\alpha)^7)$ and the analogous $P_{3,3,N}$. To control the minor arcs, the paper reduces zero relations among shifted recurrence terms to terminal conditions on a finite carry automaton: with reversed signed digits $c_i$ and carries $x_i=x_{i-1}+x_{i-2}+c_i$, the identities $x_J+x_{J-1}=\sum c_jF_{j+1}$ and $x_J+2x_{J-1}=\sum c_jL_j$ turn recurrence relations into path counts through a $361$-state transition matrix; a certified positive-vector bound gives the needed large-value estimates. On the major arcs, an imported prime-pair package (Proposition 3.1) supplies a uniform asymptotic for the weighted prime sum, with exceptional moduli handled by a recurrence analogue of the classical density argument.
Extended reading notes
Core claim
On the paper's own terms, the discovery is that the method for representing large integers as two primes plus a bounded number of sparse-sequence terms extends from first-order lacunary sequences to the second-order Fibonacci and Lucas recurrences. The Fibonacci numbers are $F_0=0,F_1=1,F_{n+1}=F_n+F_{n-1}$, and the Lucas numbers obey the same recurrence with $L_0=2,L_1=1$. The proof defines a prime sum $S(\alpha)$ over primes in $(N^{0.99},N]$, recurrence sums $T_F,T_L$ over indices up to $J=\lfloor\vartheta\log N/\log\varphi\rfloor$ with $\vartheta=1999/2000$, and representation integrals such as $R_e(N)=\int_0^1 S(\alpha)^2A(\alpha)^2B(\alpha)^4e(-N\alpha)\,d\alpha$, where $A$ and $B$ collect the even and odd Fibonacci terms by index modulo $3$. Positivity of the integer coefficients in these integrals means that a positive value of $R_e(N)$ is exactly a representation. The paper proves that the major arcs contribute more than the minor arcs by a positive margin in each of the three cases.
Load-bearing premise
The whole proof assumes Proposition 3.1, an imported uniform major-arc asymptotic stating that for all even $m\in[N/2,N]$ and all even differences $h$ with $|h|\le N^{\vartheta}/50$, the weighted prime-pair count is equal to its singular-series main term up to an arbitrarily small relative error outside a finite set of exceptional squarefree moduli; the preprint's sketch does not display the control of those exceptional conductors, and the numerical constants rest on companion certificates that are not included. If that uniformity fails by more than the declared $\omega=10^{-4}$, the final positive margins are not guaranteed.
Editorial extensions
If this is right
- Theorem 1.1 has the corollary, already noted in the paper, that $\{p+F_a+F_b+F_c:\ p\text{ prime},\ a,b,c\ge1\}$ is an asymptotic basis of order two, so every sufficiently large integer is a sum of two elements of that set.
- The parity structure forces even $N$ to be represented with two even and four odd Fibonacci terms, and odd $N$ with one even and five odd terms, so the Fibonacci parity pattern is built into the representation rather than being an extra freedom.
- For the Lucas and mixed problems, the analogous parity projectors show that the representation count can be forced to match the parity of $N$ while keeping all surviving coefficients nonnegative.
- If the displayed constants are correct, the smallest certified gap, about $0.000731998537761$ in the even six-Fibonacci case, is still positive, so the three theorems are established with a definite numerical margin rather than by an asymptotic tie.
Reading between the lines
- I infer that the same finite-carry method should apply to other second-order recurrences whose terms satisfy the same modulo periods and carry-confinement bounds, so analogous statements may hold for other companion recurrence sequences; the paper does not claim this.
- Because the even six-Fibonacci case has the smallest margin, a fully explicit effective version would amount to auditing the unprinted numerical certificates from Sections 4, 7, and 8, and that is the natural place to stress-test the proof.
- One can also read the exceptional-modulus lemma as a general recipe: any sequence whose residues modulo squarefree moduli are sufficiently evenly distributed can be appended to a two-prime problem without changing the main term, which suggests a broad family of additions beyond Fibonacci and Lucas numbers.
Original abstract
Via the circle method, we deduce that all sufficiently large positive integers are sums of two primes and six positive Fibonacci numbers, and are also sums of two primes and seven Lucas numbers. We also prove that each sufficiently large positive integer can be written as a sum of two primes, three Fibonacci numbers, and three Lucas numbers.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Via the circle method, we deduce that all sufficiently large positive integers are sums of two primes and six positive Fibonacci numbers, and are also sums of two primes and seven Lucas numbers. We also prove that each sufficiently large positive integer can be written as a sum of two primes, three Fibonacci numbers, and three Lucas numbers.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Via the circle method, we deduce that all sufficiently large positive integers are sums of two primes and six positive Fibonacci numbers, and are also sums of two primes and seven Lucas numbers. | result 0.98 | 0.95 | 0.71 | asserted 1.00 |
| s1 | We also prove that each sufficiently large positive integer can be written as a sum of two primes, three Fibonacci numbers, and three Lucas numbers. | result 0.98 | 0.95 | 0.20 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
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- relations to other papers
- P abstract
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References
How the literature uses it
What this paper does with the arXiv papers it cites
What it cites · 4 sentences without a confident label
Uses the method or tool of 1 paper
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For the Goldbach formula, use [ref] .
A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach and Generalized Twin Prime Problems · The two major-arc scales · 0.99
Uses data or values from 2 papers
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For a nonzero integer $h$, we set [equation] Johnston and Trudgian [ref] proved that uniformly for $0<h<N$ we have [equation] Multiplication by $logp_1logp_2≤(logN)^2$ and the normalization in [ref] yields that for every nonzero even integer $h$ with $|h|<N$ we have [equation] We reserve $0.0003$ and use [equation]
An update on the Linnik--Goldbach problem · The shifted-prime coefficient · 0.85
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For the major-arc parameter $P>1$, set [equation] Pintz [ref] permits the choice $P_0=N^0.41$ and $_0=0.005$, and hence a cutoff [equation] Pintz and Ruzsa [ref] permits $P_0=N^2/5$ and $_0=0.001$, and hence [equation] Define [equation] The numerical ranges give the pointwise inclusion $⊂$.
A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach and Generalized Twin Prime Problems · The two major-arc scales · 0.92
Discussion
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